How do you integrate #int xsqrt(1-x^2)dx# from [0,1]?

Answer 1

#int_0^1 xsqrt(1-x^2)dx = 1/3#

Substitute:

#t = 1-x^2# #dt = -2xdx#
#int_0^1 xsqrt(1-x^2)dx = -1/2int_1^0 sqrt(t) dt = 1/2int_0^1 sqrt(t) dt =#
# = 1/3 t^(3/2) |_(x=0)^(x=1) =1/3#
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Answer 2

#1/3#

If one recognises that the outside of the sqrt is a function of the sqrt differentiated we can proceed by inspection.

#int_0^1x(1-x^2)^(1/2)dx#
guess#" " y=(1-x^2)^(3/2)#
#d/(dx)((1-x^2)^(3/2))=3/2xx(-2x)(1-2x)^(1/2)#
#=-3(1-2x)^(1/2)#
#:.int_0^1x(1-x^2)^(1/2)dx=-1/3[(1-2x)^(3/2)]_0^1#
#-1/3{[(1-2x)^(3/2)]^1-[(1-2x)^(3/2)]_0}#
#-1/3((0)-1)#
#=1/3#
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Answer 3

To integrate the function ( \int x \sqrt{1-x^2} , dx ) over the interval ([0,1]), you can use the trigonometric substitution method. Let's substitute ( x = \sin \theta ) and ( dx = \cos \theta , d\theta ). After substitution, the integral becomes ( \int \sin \theta \sqrt{1-\sin^2 \theta} \cos \theta , d\theta ).

Now, ( \sqrt{1-\sin^2 \theta} = \cos \theta ), so the integral simplifies to ( \int \sin^2 \theta , d\theta ) over the interval ([0, \frac{\pi}{2}]).

Using the identity ( \sin^2 \theta = \frac{1 - \cos 2\theta}{2} ), the integral further simplifies to ( \frac{1}{2} \int (1 - \cos 2\theta) , d\theta ).

Integrating term by term, you get ( \frac{1}{2} (\theta - \frac{1}{2} \sin 2\theta) + C ), where ( C ) is the constant of integration.

Now substitute back ( x = \sin \theta ) and ( dx = \cos \theta , d\theta ), and apply the limits ([0,1]). After substituting and simplifying, you'll get the final result for the definite integral over the interval ([0,1]).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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